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0256. Now note that as f j = 0 for j = 4, 5, 6, . . 1905. 7 Notes and references Further details of the distributions discussed in this chapter, including a discussion of how to fit parameters to these distributions, can be found in Hogg and Klugman (1984). See also Klugman et al. (1998). 3 was derived by De Pril (1985), and a very elegant proof of the result can be found in his paper. 8 Exercises 1. A random variable X has a logarithmic distribution with parameter θ , where 0 < θ < 1, if its probability function is Pr(X = x) = −1 θx log(1 − θ) x for x = 1, 2, 3, .

72, so that the individual would be prepared to pay a premium of 208. 2 Quadratic A utility function of the form u(x) = x − βx 2 , for x < 1/(2β) and β > 0, is called a quadratic utility function. The use of this type of utility function is restricted by the constraint x < 1/(2β), which is required to ensure that u (x) > 0. Thus, we cannot apply the function to problems under which random outcomes are distributed on (−∞, ∞). As indicated in the previous section, decisions made using a quadratic utility function depend only on the first two moments of the random outcomes, as illustrated in the following examples.

Thus, the insurer requires a premium that is at least equal to the expected loss, and so an insurance contract is feasible when P ≥ . 5 Types of utility function It is possible to construct a utility function by assigning different values to different levels of wealth. For example, an individual might set u(0) = 0, u(10) = 5, u(20) = 8, and so on. Clearly it is more practical to assign values through a suitable mathematical function. Therefore, we now consider some mathematical functions which may be regarded as having suitable forms to be utility functions.

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